In today's digital era, a wide variety of signals—such as video and audio signals—are digitalized. A few examples of products that use digital signals include digital TV, Bluetooth headphones, DVD players, WAP mobile phone handsets, etc. To ensure that the signals used in digital products may be read properly, enabling the products to present high-definition video and audio even when the signals have been transmitted over long distances, the signals are typically encoded and decoded. However, since transmission media and channels are easily corrupted by interference during data access and transmission, error detection and correction become more and more significant. Generally, error-correcting codes are widely used for enhancing reliability of data access and transmission. In the error-correcting codes, application of a cyclic code is not uncommon.
A finite field (also known as a Galois field) is a field composed of a finite number of elements. The number of elements in the field called the order or cardinality of the field. This number is always of the form pm, where p is a prime number and m is a positive integer. A Galois field of order q=pm will be designated either as GF (pm) or as Fq in the following. These symbols GF (pm) and Fq are fully synonymous. A polynomial over an arbitrary field (including a finite field) will be designated as p(x), or a similar symbol. A element in which the polynomial is to be evaluated will be designated by lower-case Greek letters such as α, β or γ, in the following. The definitions and properties of finite fields are described in many standard textbooks of mathematics, and reference is made to such standard textbooks for details.
In conventional cases, the error correction method of a cyclic code involves the use of an algebraic decoding method to eliminate syndromes from among the Newton's identities so as to obtain the error polynomial coefficient, which in turn may be used to obtain the error polynomial. However, with the increasingly higher requirements of communication in post-Internet era, the length and the kinds of cyclic codes increases, it becomes increasingly difficult for the high order equations produced when using an algebraic method to find a solution over a Galois field, making it difficult to obtain the error polynomial.
To solve the problems discussed above, which typically arises in the realm of computer networks and post-Internet era communication need, the present disclosure provides an error correction system applicable to all cyclic codes.